1D or 2D goniometry method of diffuse sources

ABSTRACT

A goniometry method for one or several diffuse (or distributed) sources is disclosed. The sources or sources having one or more give directions and a diffusion cone. The sources are received by an array of several sensors. The method breaks down the diffusion cone into a finite number L of diffusers. A diffuser has the parameters (θ mp , δθ mpi , Δ mp , δΔ mpi ), associated with it. Directing vectors a(θ mp +δθ mpi , Δ mp +δΔ mpi ) associated with the L diffusers are combined to obtain a vector (D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α)) dependent on at least one of the incidence and deflection parameters (θ, Δ, δθ, δΔ) and on the combination vector α. A MUSIC-type criterion or other goniometry algorithm is applied to the vectors D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α) obtained in order to determine at least one of the incidence parameters θ mp , Δ mp , δθ mp , δΔ mp  of the associated diffusion cone.

RELATED APPLICATIONS

The present Application is based on International Application No.PCT/EP2005/050430, filed on Feb. 1, 2005, which in turn corresponds toFrench Application No. 02 11072 filed on Sep. 6, 2002, and priority ishereby claimed under 35 USC §119 based on these applications. Each ofthese applications are hereby incorporated by reference in theirentirety into the present application.

FIELD OF THE INVENTION

The present invention relates to a goniometry method for one or morediffuse, or “distributed”, radiofrequency sources, the source of givendirection being considered by the receivers as a diffusion cone with acertain width and an average incidence.

BACKGROUND OF THE INVENTION

A distributed source is defined notably as a source which is propagatedthrough a continuum of diffusers.

The invention makes it possible notably to locate, in angles and/or inazimuth, one or more distributed radio frequency sources. The object is,for example, to determine the incidence of the centers of the diffusioncones and their widths.

The goniometry is produced either in one dimension, 1D, where theincidences are parameterized by the azimuth, or in two dimensions, 2D,where the incidence depends on both azimuth and elevation parameters.

It applies, for example, for decorrelated or partially decorrelatedcoherent signals originating from diffusers.

FIG. 1 diagrammatically represents the example of diffusion of the wavefrom cell phone M through a layer of snow N_(G), for example to thereceivers Ci of the reception system of an airplane A. The cone, calleddiffusion cone, has a certain width and an average incidence. The snowparticles N_(G) act as diffusers.

In the field of antenna processing, a multiple-antenna system receivesone or more radiocommunication transmitters. The antenna processingtherefore uses the signals originating from multiple sensors. In anelectromagnetic context, the sensors are antennas. FIG. 2 shows how anyantenna processing system consists of an array 1 with several antennas 2(or individual sensors) receiving the multiple paths from multipleradiofrequency transmitters 3, 4, from different incidence angles and anantenna processing device 5. The term “source” is defined as a multiplepath from a transmitter. The antennas of the array receive the sourceswith a phase and an amplitude dependent on their incidence angle and onthe positioning of the antennas. The incidence angles can beparameterized, either in 1D azimuth-wise θ_(m), or in 2D, azimuth-wiseθ_(m) and elevation-wise Δ_(m). FIG. 3 shows that a goniometry isobtained in 1D when the waves from the transmitters are propagated inone and the same plane and a 2D goniometry must be applied in othercases. This plane P can be that of the array of antennas where theelevation angle is zero.

The main objective of the antenna processing techniques is to exploitthe space diversity, namely, the use of the spatial position of theantennas of the array to make better use of the incidence and distancedivergences of the sources. More particularly, the objective of thegoniometry or the locating of radiofrequency sources is to estimate theincidence angles of the transmitters from an array of antennas.

Conventionally, the goniometry algorithms such as MUSIC described, forexample, in reference [1] (the list of references is appended) assumethat each transmitter is propagated according to a discrete number ofsources to the listening receivers. The wave is propagated either with adirect path or along a discrete number of multiple paths. In FIG. 2, thefirst transmitter referenced 3 is propagated along two paths ofincidences θ₁₁ and θ₁₂ and the second transmitter referenced 4 along adirect path of incidence θ₂. To estimate the incidences of all of thesediscrete sources, their number must be strictly less than the number ofsensors. For sources that have diffusion cones of non-zero width, thegoniometry methods described in document [1] are degraded because of theinadequacy of the signal model.

References [2] [3] [4] propose solutions for the goniometry ofdistributed sources. However, the proposed goniometry algorithms are inazimuth only: 1D. Also, the time signals of the diffusers originatingfrom one and the same cone are considered to be either coherent inreferences [2] and [3], or incoherent in references [3] [4]. Physically,the signals of the diffusers are coherent when they are not temporallyshifted and have no Doppler shift. Conversely, these signals areincoherent when they are strongly shifted in time or when they have asignificant Doppler shift. The time shift of the diffusers depends onthe length of the path that the waves follow through the diffusers andthe Doppler depends on the speed of movement of the transmitter or ofthe receivers. These comments show how references [2] [3] [4] do nothandle the more common intermediate case of diffusers with partiallycorrelated signals. Also, the algorithms [2] [4] strongly depend on an“a priori” concerning the probability density of the diffusion conesangle-wise. It is then sufficient for these densities to be slightlydifferent from the “a priori” for the algorithms [2] [4] no longer to besuitable.

SUMMARY OF THE INVENTION

The subject of the invention concerns notably distributed sources whichare received by the listening system in a so-called diffusion conehaving a certain width and an average incidence as described for examplein FIG. 1.

In this document, the word “source” denotes a multiple-path by diffusionfrom a transmitter, the source being seen by the receivers in adiffusion cone of a certain width and an average incidence. The averageincidence is defined notably by the direction of the source.

The invention relates to a goniometry method for one or several diffusesources of given directions, the source or sources being characterizedby one or more given directions and by a diffusion cone. It ischaracterized in that it comprises at least the following steps:

-   a) breaking down the diffusion cone into a finite number L of    diffusers, a diffuser having the parameters (θ_(mp), δθ_(mpi),    Δ_(mp), δΔ_(mpi)), associated with it,-   b) combining the directing vectors a(θ_(mp)+δθ_(mpi),    Δ_(mp)+δΔ_(mpi)) associated with the L diffusers to obtain a vector    (D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α)) dependent on at least    one of the incidence and deflection parameters (θ, Δ, δθ, δΔ) and on    the combination vector α,-   c) applying a MUSIC-type criterion or any other goniometry algorithm    to the vectors D(θ, Δ, δθ, δΔ) α or U(θ, Δ) β(δθ, δΔ, α) obtained in    the step b) in order to determine at least one of the incidence    parameters θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp) of the associated    diffusion cone.

The minimizing step is, for example, performed on the matrix D(θ, Δ, δθ,δΔ) and implemented according to the parameters θ, Δ, δθ, δΔ.

The minimizing step can be performed on the matrix D_(s)(θ, Δ, δθ, δΔ)where the parameters δθ and/or δΔ are replaced by their opposites.

According to a variant of embodiment, the algorithm comprises a step oflimited development of the directing vectors about the central incidenceof the cone in order to separate the incidences (θ, Δ) and thedeflections δθ, δΔ and in that the minimizing step is performedaccording to the parameters (θ, Δ) on a matrix U(θ, Δ) dependent on theincidences in order to determine the parameters θ_(mp), Δ_(mp)minimizing the criterion, then secondly to determine the deflectionparameters δθ_(mp), δΔ_(mp) from the parameters θ_(mp), Δ_(mp).

The minimizing step is, for example, performed on the matrix U_(s)(θ, Δ)dependent on U(θ, Δ).

The matrix D(θ, δθ) can be dependent only on the azimuth angle θ and onthe deflection vector δθ of this angle.

The minimizing step is, for example, performed on the matrix D_(s)(θ,δθ), where the parameter δθ is replaced by its opposite.

The method can include a step of limited development of the vectors ofthe matrix D(θ, δθ), the minimizing step being performed on a matrixU(θ) in order to determine the incidence angle parameters θ_(mp) and,from these parameters, the angle offset parameters δθ_(mp).

The minimizing step is performed on the matrix U_(s)(θ) dependent onU(θ).

The object of the invention has notably the following advantages:

-   -   producing a goniometry in azimuth and/or in azimuth-elevation,    -   reducing the calculation cost of the method by using a limited        development of the directing vectors,    -   taking into account any type of diffusers, notably        partially-correlated diffusers.

Still other objects and advantages of the present invention will becomereadily apparent to those skilled in the art from the following detaileddescription, wherein the preferred embodiments of the invention areshown and described, simply by way of illustration of the best modecontemplated of carrying out the invention. As will be realized, theinvention is capable of other and different embodiments, and its severaldetails are capable of modifications in various obvious aspects, allwithout departing from the invention. Accordingly, the drawings anddescription thereof are to be regarded as illustrative in nature, andnot as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not bylimitation, in the figures of the accompanying drawings, whereinelements having the same reference numeral designations represent likeelements throughout and wherein:

FIG. 1, a representation of the diffusion of a wave from a cell phonethrough a layer of snow,

FIG. 2, an exemplary architecture of an antenna processing system,

FIG. 3, an exemplary 2D goniometry in azimuth-elevation,

FIG. 4, a diagram of the steps of the first variant of the goniometry ofdistributed sources in azimuth and elevation,

FIG. 5, a diagram of a variant of FIG. 4, taking account of the angularsymmetry of the diffusion cones,

FIG. 6, a second variant of the goniometry method for the distributedsources in azimuth and bearing,

FIG. 7, the symmetrical version of the variant of FIG. 6,

FIG. 8, the steps of the first variant of the goniometry in azimuth ofthe distributed sources,

FIGS. 9 and 10, results of goniometry in azimuth of one or moredistributed sources associated with the first variant of the algorithm,

FIG. 11, the symmetrical variant of FIG. 8,

FIG. 12, the steps associated with a second variant of the goniometry inazimuth method,

FIGS. 13 and 14, two results of goniometry in azimuth of distributedsources,

FIG. 15, a diagram of the symmetrical version of the second goniometryin azimuth variant.

DETAILED DESCRIPTION OF THE INVENTION

In order to better understand the method according to the invention, thedescription that follows is given, as an illustration and in anonlimiting way, in the context of the diffusion of the wave from a cellphone through a layer of snow to the receivers on an airplane, forexample represented in FIG. 1. The snow particles act as diffusers.

In this example, a diffuse or distributed source is characterized, forexample, by a direction and a diffusion cone.

Before detailing the exemplary embodiment, a few reminders are giventhat may be helpful in understanding the method according to theinvention.

General Case

In the presence of M transmitters being propagated along P_(m)non-distributed multiple paths of incidences (θ_(mp), Δ_(mp)) arrivingat an array consisting of N sensors, the observation vector x(t) belowis received at the output of the sensors:

$\begin{matrix}\begin{matrix}{{x(t)} = \begin{bmatrix}{x_{1}(t)} \\\vdots \\{x_{N}(t)}\end{bmatrix}} \\{= {{\sum\limits_{m = 1}^{M}\;{\sum\limits_{p = 1}^{P_{m}}{\rho_{m\; p}{a\left( {\theta_{m\; p},\Delta_{m\; p}} \right)}{s_{m}\left( {t - \tau_{m\; p}} \right)}{\mathbb{e}}^{j\; 2\pi\;{fmp}\; t}}}} + {b(t)}}}\end{matrix} & (1)\end{matrix}$where x_(n)(t) is the signal received on the nth sensor, a(θ,Δ) is theresponse from the array of sensors to a source of incidence θ, Δ,s_(m)(t) is the signal transmitted by the mth transmitter, τ_(mp),f_(mp) and ρ_(mp) are respectively the delay, the Doppler shift and theattenuation of the pth multiple path of the mth transmitter and x(t) isthe additive noise.

To determine the M_(T)=P₁+ . . . +P_(M) incidences (θ_(mp), Δ_(mp)), theMUSIC method [1] seeks the M_(T) minima ({circumflex over(θ)}_(mp),{circumflex over (Δ)}_(mp)) that cancel the followingpseudo-spectrum:

$\begin{matrix}{{{J_{MUSIC}\left( {\theta,\Delta} \right)} = \frac{{a^{H}\left( {\theta,\Delta} \right)}\Pi_{b}{a\left( {\theta,\Delta} \right)}}{{a^{H}\left( {\theta,\Delta} \right)}{a\left( {\theta,\Delta} \right)}}},} & (2)\end{matrix}$where the matrix Π_(b) depends on the (N−M_(T)) natural vectors e_(MT+i)(1≦i≦N−M_(T)) associated with the lowest natural values of thecovariance matrix R_(xx)=E[x(t) x(t)^(H)]: Π_(b)=E_(b)E_(b) ^(H) whereE_(b)=[e_(MT+1) . . . e_(N)]. It will also be noted that u^(H) is theconjugate transpose of the vector u. The MUSIC method is based on thefact that the M_(T) natural vectors e_(i) (1≦i≦M_(T)) associated withthe highest natural values generate the space defined by the M_(T)directing vectors a(θ_(mp),Δ_(mp)) of the sources such as:

$\begin{matrix}{{e_{i} = {\sum\limits_{m = 1}^{M}\;{\sum\limits_{p = 1}^{P_{m}}{\alpha_{mpi}{a\left( {\theta_{m\; p},\Delta_{m\; p}} \right)}}}}},} & (3)\end{matrix}$and that the vectors e_(i) are orthogonal to the vectors of the noisespace e_(i+MT).

In the presence of M transmitters being propagated along P_(m)distributed multiple paths, the following observation vector x(t) isobtained:

$\begin{matrix}{{x(t)} = {{\sum\limits_{m = 1}^{M}\;{\sum\limits_{p = 1}^{P_{m}}{x_{m\; p}(t)}}} + {b(t)}}} & (4)\end{matrix}$such that

$\begin{matrix}{{x_{m\; p}(t)} = {\int_{\theta_{m\; p} - {\delta\;\theta_{m\; p}}}^{\theta_{m\; p} + {\delta\;\theta_{m\; p}}}{\int_{\Delta_{m\; p} - {\delta\;\Delta_{m\; p}}}^{\Delta_{m\; p} + {\delta\;\Delta_{m\; p}}}{{\rho\left( {\theta,\Delta} \right)}{a\left( {\theta,\Delta} \right)}{s_{m}\left( {t - {\tau\left( {\theta,\Delta} \right)}} \right)}\ {\mathbb{e}}^{j\; 2\;\pi\;{f{({\theta,\Delta})}}t}{\mathbb{d}\theta}\ {\mathbb{d}\Delta}}}}} & \;\end{matrix}$where (θ_(mp),Δ_(mp)) and (δθ_(mp),δΔ_(mp)) respectively denote thecenter and the width of the diffusion cone associated with the pthmultiple path of the mth transmitter. The parameters τ(θ, Δ), f(θ, Δ)and ρ(θ, Δ) are respectively the delay, the Doppler shift and theattenuation of the diffuser of incidence (θ, Δ). In the presence ofcoherent diffusers, the delay τ(θ, Δ) and the Doppler shift f(θ, Δ) arezero.Theory of the Method According to the Invention

The invention is based notably on a breakdown of a diffusion cone into afinite number of diffusers. Using L to denote the number of diffusers ofa source, the expression [4] can be rewritten as the followingexpression [5]:

${x_{m\; p}(t)} = {\sum\limits_{i = 1}^{L}{\rho_{i}{a\left( {{\theta_{m\; p} + {\delta\;\theta_{mpi}}},{\Delta_{m\; p} + {\delta\;\Delta_{mpi}}}} \right)}{s_{m}\left( {t - \tau_{m\; p} - {\delta\;\tau_{mpi}}} \right)}{\mathbb{e}}^{j\; 2\;{\pi{({{fmp} + {\delta\;{fmpi}}})}}t}}}$

The expression (5) makes it possible to bring things back to the modelof discrete sources (diffusers) of the expression [1] by consideringthat the individual source is the diffuser of incidence(θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)) associated with the ith diffuser ofthe pth multiple path of the mth transmitter. In these conditions, thesignal space of the covariance matrix R_(xx)=E[x(t) x(t)^(H)] isgenerated by the vectors a(θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)). By using Kto denote the rank of the covariance matrix R_(xx), it can be deducedfrom this that its natural vectors e_(i) (1≦i≦K) associated with thehighest natural values satisfy, according to [3], the followingexpression:

$\begin{matrix}{e_{i} = {{\sum\limits_{m = 1}^{M}\;{\sum\limits_{p = 1}^{P_{m}}{{c\left( {\theta_{m\; p},\Delta_{m\; p},{\delta\;\theta_{m\; p}},{\delta\;\Delta_{m\; p}},\alpha_{m\; p}^{i}} \right)}\mspace{14mu}{for}\mspace{14mu} 1}}} \leq i \leq K}} & (6)\end{matrix}$such that

${c\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}{\alpha_{j}{a\left( {{\theta + {\delta\;\theta_{j}}},{\Delta + {\delta\;\Delta_{j}}}} \right)}}}$with

${{\delta\;\theta} = \begin{bmatrix}{\delta\;\theta_{1}} \\\vdots \\{\delta\;\theta_{L}}\end{bmatrix}},{{\delta\;\Delta} = {{\begin{bmatrix}{\delta\;\Delta_{1}} \\\vdots \\{\delta\;\Delta_{L}}\end{bmatrix}\mspace{14mu}{and}{\mspace{11mu}\;}\alpha} = \begin{bmatrix}\alpha_{1} \\\vdots \\\alpha_{L}\end{bmatrix}}}$

In the presence of coherent diffusers where δτ_(mpi)=0 and δf_(mpi)=0,it should be noted that the rank of the covariance matrix satisfies:K=M_(T)=P₁+ . . . +P_(M). In the general case of partially-correlateddiffusers where δτ_(mpi)≠0 and δf_(mpi)≠0, this rank satisfiesK≧M_(T)=P₁+ . . . +P_(M). In the present invention, it is assumed thatc(θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp), α_(mp) ^(i)) is one of the directingvectors associated with the pth multiple path of the mth transmitter andthat the unknown parameters are the average incidence (θ_(mp),Δ_(mp)),the angle differences of the diffusers (δθ_(mp), δΔ_(mp)) and one of thevectors α_(mp) ^(i).

Case of Goniometry in Azimuth and in Elevation

1st Variant

FIG. 4 diagrammatically represents the steps implemented according to afirst variant of embodiment of the method.

To sum up, the diffusion cone is broken down into L individual diffusers(equation [5]), the different directing vectors a(θ_(mp)+δθ_(mpi),Δ_(mp)+δΔ_(mpi)) are combined, which causes a vector D(θ, Δ, δθ, δΔ) αto be obtained, to which is applied a MUSIC-type or goniometry criterionin order to obtain the four parameters θ_(mp), Δ_(mp), δθ_(mp), δΔ_(mp)which minimize this criterion (the MUSIC criterion is applied to avector resulting from the linear combination of the different directingvectors).

To determine these parameters with a MUSIC-type algorithm [1], it isessential, according to equations [2] and [6], to find the minima({circumflex over (θ)}_(mp),{circumflex over (Δ)}_(mp),δ{circumflex over(θ)}_(mp),{circumflex over (α)}_(mp) ^(i)) which cancel the followingpseudo-spectrum:

$\begin{matrix}{{{J_{MUSIC\_ diff}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = \frac{\begin{matrix}{c^{H}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} \\{\Pi_{b}{c\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}}\end{matrix}}{\begin{matrix}{c^{H}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} \\{c\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}\end{matrix}}},} & (7)\end{matrix}$Where the matrix Π_(b) depends on the (N−K) eigenvectors e_(MT+i)(1≦i≦N−K) associated with the lowest natural values of the covariancematrix R_(xx)=E[x(t) x(t)^(H)]: Π_(b)=E_(b) E_(b) ^(H) whereE_(b)=[e_(K+1) . . . e_(N)]. Noting, according to the expression [6],that the vector c(θ, Δ, δθ, δΔ, α) can be written in the following form:c(θ,Δ,δθ,δΔ,α)=D(θ,Δ,δθ,δΔ)α,  (8)

-   -   with D(θ, Δ, δθ, δΔ)=[a(θ+δθ₁, Δ+δΔ₁) . . . a(θ+δθ_(L),        Δ+δΔ_(L))],        it is possible to deduce from this that the criterion J_(MUSIC)        _(—) _(diff)(θ, Δ, δθ, δΔ, α) becomes:

$\begin{matrix}{{{J_{MUSIC\_ diff}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = \frac{\alpha^{H}{Q_{1}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta}} \right)}\alpha}{\alpha^{H}{Q_{2}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta}} \right)}\alpha}},} & (9)\end{matrix}$with Q₁(θ, Δ, δθ, δΔ)=D(θ, Δ, δθ, δΔ)^(H) Π_(b) D(Oθ, Δ, δθ, δΔ),and Q₂(θ, Δ, δθ, δΔ)=D(θ, Δ, δθ, δΔ)^(H) D(θ, Δ, δθ, δΔ),

The technique will firstly consist in minimizing the criterion J_(MUSIC)_(—) _(diff)(θ, Δ, δθ, δΔ, α) with α. According to the techniquedescribed in reference [2], for example, the criterion J_(min) _(—)_(diff)(θ, Δ, δθ, δΔ) below is obtained:J _(min) _(—) _(diff)(θ,Δ,δθ,δΔ)=λ_(min) {Q ₁(θ,Δ,δθ,δΔ)Q₂(θ,Δ,δθ,δΔ)⁻¹}  (10)where λ_(min)(Q) denotes the minimum natural value of the matrix Q.Noting that the criterion J_(min) _(—) _(diff)(θ, Δ, δθ, δΔ) should becancelled out for the quadruplets of parameters(θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) and that det(AB⁻¹)=det(A)/det(B), it canbe deduced from this that the quadruplets of parameters(θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) also cancel the following criterion:J _(diffision)(θ,Δ,δθ,δΔ)=det(Q ₁(θ,Δ,δθ,δΔ))/det(Q ₂(θ,Δ,δθ,δΔ),  (11)where det(Q) denotes the determinant of the matrix Q. The M_(T)quadruplets of parameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) which minimizethe criterion J_(diffusion)(θ, Δ, δθ, δΔ) are therefore sought.

FIG. 5 represents the steps of a variant of embodiment taking account ofthe symmetry of the incidence solution.

Indeed, if (θ,Δ,δθ,δΔ) is the solution, the same applies for(θ,Δ,−δθ,δΔ) (θ,Δ,δθ,−δΔ) (θ,Δ,−δθ,−δΔ). From this comment, it ispossible to deduce that:c(θ,Δ,δθ,δΔ,α)^(H) E _(b)=0,c(θ,Δ,−δθ,δΔ,α)^(H) E _(b)=0,c(θ,Δ,δθ,−δΔ,α)^(H) E _(b)=0,c(θ,Δ,−δθ,−δΔ,α)^(H) E _(b)=0  (11-1)Where the matrix E_(b) depends on the (N−K) eigenvectors e_(MT+i)(1≦i≦N−K) associated with the lowest eigenvalues of the covariancematrix R_(xx)=E[x(t) x(t)^(H)] such that: E_(b)=[e_(K+1) . . . e_(N)].From the expression (11-1) it can be deduced that, to estimate theparameters (θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)), it is necessary to find theminima that cancel the following pseudo-spectrum:

$\begin{matrix}{{{{J_{{MUSIC\_ diff}{\_ sym}}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = \frac{\begin{matrix}{c_{s}^{H}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} \\{\Pi_{bs}{c_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}}\end{matrix}}{\begin{matrix}{c_{s}^{H}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} \\{c_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}\end{matrix}}},{{c_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = {\begin{bmatrix}{c\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} \\{c\left( {\theta,\Delta,{{- \delta}\;\theta},{\delta\;\Delta},\alpha} \right)} \\{c\left( {\theta,\Delta,{\delta\;\theta},{{- \delta}\;\Delta},\alpha} \right)} \\{c\left( {\theta,\Delta,{{- \delta}\;\theta},{{- \delta}\;\Delta},\alpha} \right)}\end{bmatrix}\mspace{14mu}{and}}}}{\Pi_{bs} = {\frac{1}{4}E_{bs}E_{bs}^{H}}}{where}{E_{bs} = \begin{bmatrix}E_{b} \\E_{b} \\E_{b} \\E_{b}\end{bmatrix}}} & \left( {11\text{-}2} \right)\end{matrix}$According to the expression (8), the vector c_(s)(θ, Δ, δθ, δΔ, α) canbe written as follows:

$\begin{matrix}{{{{c_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = {{D_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta}} \right)}\alpha}},{with}}{{{D_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta}} \right)} = \begin{bmatrix}{D\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta}} \right)} \\{D\left( {\theta,\Delta,{{- \delta}\;\theta},{\delta\;\Delta}} \right)} \\{D\left( {\theta,\Delta,{\delta\;\theta},{{- \delta}\;\Delta}} \right)} \\{D\left( {\theta,\Delta,{{- \delta}\;\theta},{{- \delta}\;\Delta}} \right)}\end{bmatrix}},}} & \left( {11\text{-}3} \right)\end{matrix}$

The minimizing of J_(MUSIC) _(—) _(diff) _(—) _(sym)(θ, Δ, δθ, δΔ, α)relative to α will lead to the criterion J_(diffusion) _(—) _(sym)(θ, Δ,δθ, δΔ). To obtain J_(diffusion) _(—) _(sym)(θ, Δ, δθ, δΔ), all that isneeded is to replace in the expressions (9) (11), D(θ, Δ, δθ, δΔ) withits symmetrical correspondent D_(s)(θ, Δ, δθ, δΔ) and Π_(b) with Π_(bs).The following is thus obtained:J _(diffusion-sym)(θ,Δ,δθ,δΔ)=det(Q _(1s)(θ,Δ,δθ,δΔ)/det(Q_(2s)(θ,Δ,δθ,δΔ)),  (11-4)with Q_(1s)(θ, Δ, δθ, δΔ)=D_(s)(θ, Δ, δθ, δΔ)^(H) Π_(bs) D_(s)(θ, Δ, δθ,δΔ),and Q_(2s)(θ, Δ, δθ, δΔ)=D_(s)(θ, Δ, δθ, δΔ)^(H) D_(s)(θ, Δ, δθ, δΔ),

Therefore, the MT quadruplets of parameters(θ_(mp),Δ_(mp),δθ_(mp),δΔ_(mp)) which minimize the criterionJ_(diffusion-sym)(θ, Δ, δθ, δΔ) are sought.

2nd Variant

FIG. 6 represents a second variant of the goniometry of the diffusesources in azimuth and elevation that offer notably the advantage ofreducing the calculation costs.

The first variant of the goniometry of the sources involves calculatinga pseudo-spectrum J_(diffusion) dependent on four parameters (θ, Δ, δθ,δΔ), two of which are vectors of length L. The objective of the secondvariant is to reduce this number of parameters by performing the limiteddevelopment along directing vectors about a central incidence (θ, Δ)corresponding to the center of the diffusion cone:

$\begin{matrix}{{a\left( {{\theta + {\delta\;\theta_{i}}},{\Delta + {\delta\;\Delta_{i}}}} \right)} = {{a\left( {\theta,\Delta} \right)} + {\delta\;\theta_{i}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}} + {\delta\;\Delta_{i}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}} + {\ldots\mspace{14mu}{etc}}}} & (12)\end{matrix}$where ∂(a(θ, Δ))^(n)/∂θ^(n−p)∂Δ^(p) denotes an nth derivative of thedirecting vector a(θ,Δ). This corresponds to a limited development aboutthe central incidence (change of base of the linear combination)according to the derivatives of the directing vectors dependent on thecentral incidence of the cone. From this last expression, it is possibleto separate the incidences (θ,Δ) and the deflections (δθ,δΔ) as follows:

$\begin{matrix}{{{a\left( {{\theta + {\delta\;\theta_{i}}},{\Delta + {\delta\;\Delta_{i}}}} \right)} = {{U\left( {\theta,\Delta} \right)}{k\left( {{\delta\;\theta_{i}},{\delta\;\Delta_{i}}} \right)}}}{where}\text{}{{U\left( {\theta,\Delta} \right)} = \left\lbrack {{a\left( {\theta,\Delta} \right)}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu}\ldots}\mspace{14mu} \right\rbrack}{and}{{k\left( {{\delta\;\theta_{i}},{\delta\;\Delta_{i}}} \right)} = \begin{bmatrix}1 \\{\delta\;\theta_{i}} \\{\delta\;\Delta_{i}} \\\vdots\end{bmatrix}}} & (13)\end{matrix}$According to the expressions (6) (8) and (13), the vector c(θ, Δ, δθ,δΔ, α) becomes:

$\begin{matrix}{{{{c\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = {{U\left( {\theta,\Delta} \right)}{\beta\left( {{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}}},{with}}{{\beta\left( {{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}\;{\alpha_{j}{k\left( {{\delta\;\theta_{i}},{\delta\;\Delta_{j}}} \right)}}}}} & (14)\end{matrix}$

By replacing, in equation (9), D(θ, Δ, δθ, δΔ) with U(θ, Δ) and α withβ(δθ, δΔ, α), it is possible to deduce from this, according to (9) (10)(1), that to estimate the M_(T) incidences (θ_(mp), Δ_(mp)) all that isneeded is to minimize the following two-dimensional criterion:J _(diffusion) _(—) _(sym) ^(opt)(θ,Δ)=det(Q ₁ ^(opt)(θ,Δ))/det(Q ₂^(opt)(θ,Δ)),  (15)withQ₁ ^(opt)(θ, Δ)=U(θ, Δ)^(H) Π_(b) U(θ, Δ) and Q₂ ^(opt)(θ, Δ)=U(θ,Δ)^(H) U(θ, Δ),

Determining the vectors δθ_(mp) and δΔ_(mp) entails estimating thevectors β(δθ_(mp), δΔ_(mp), α). For this, all that is needed is to findthe eigenvector associated with the minimum natural value of Q₂ ^(opt)(θ_(mp), Δ_(mp))⁻¹Q₁ ^(opt)(θ_(mp), Δ_(mp)).

FIG. 7 represents a variant of the method of FIG. 6 which takes accountof the symmetry of the solutions in order to eliminate some ambiguities.For this, it is important firstly to note that, according to (13) (14):c(θ,Δ,−δθ,δΔ,α)=U ₁(θ,Δ)β(δθ,δΔ,α)

$\begin{matrix}{{with}\text{}{{{U_{1}\left( {\theta,\Delta} \right)} = \left\lbrack {{a\left( {\theta,\Delta} \right)} - {\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu}\ldots}}\mspace{14mu} \right\rbrack},{{c\left( {\theta,\Delta,{\delta\;\theta},{{- \delta}\;\Delta},\alpha} \right)} = {{U_{2}\left( {\theta,\Delta} \right)}{\beta\left( {{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}}}}{with}{{{U_{2}\left( {\theta,\Delta} \right)} = \left\lbrack {{{a\left( {\theta,\Delta} \right)}\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta}} - {\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu}\ldots}}\mspace{14mu} \right\rbrack},{{c\left( {\theta,\Delta,{{- \delta}\;\theta},{{- \delta}\;\Delta},\alpha} \right)} = {{U_{3}\left( {\theta,\Delta} \right)}{\beta\left( {{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}}}}{with}{{U_{3}\left( {\theta,\Delta} \right)} = \left\lbrack {{a\left( {\theta,\Delta} \right)} - \frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\theta} - {\frac{\partial{a\left( {\theta,\Delta} \right)}}{\partial\Delta}\mspace{14mu}\ldots}}\mspace{14mu} \right\rbrack}} & \left( {15\text{-}1} \right)\end{matrix}$From this, according to (11-2), a new expression of the vector c_(s)(θ,Δ, δθ, δΔ, α) can be deduced:

$\begin{matrix}{{{c_{s}\left( {\theta,\Delta,{\delta\;\theta},{\delta\;\Delta},\alpha} \right)} = {{U_{s}\left( {\theta,\Delta} \right)}{\beta\left( {{\delta\;\theta},{\delta\;\Delta},\alpha} \right)}\mspace{14mu}{and}}}{{U_{s}\left( {\theta,\Delta} \right)} = \begin{bmatrix}{U\left( {\theta,\Delta,\alpha} \right)} \\{U_{1}\left( {\theta,\Delta,\alpha} \right)} \\{U_{2}\left( {\theta,\Delta,\alpha} \right)} \\{U_{3}\left( {\theta,\Delta,\alpha} \right)}\end{bmatrix}}} & \left( {15\text{-}2} \right)\end{matrix}$

By replacing, in the equation (15), U_(s)(θ, Δ) with U(θ, Δ), it ispossible to deduce from this, according to (11-2) (11-4), that toestimate the M_(T) incidences (θ_(mp),Δ_(mp)) all that is needed is tominimize the following two-dimensional criterion:J _(diffusion) _(—) _(sym) ^(opt)(θ,Δ)=det(Q _(1s) ^(opt)(θ,Δ))/det(Q_(2s) ^(opt)(θ,Δ)),  (15-3)with Q_(1s) ^(opt)(θ, Δ)=U_(s)(θ, Δ)^(H) Π_(bs) U_(s)(θ, Δ) and

Q_(2s) ^(opt)(θ, Δ)=U_(s)(θ, Δ)^(H) U_(s)(θ, Δ)

Case of 1D Goniometry in Azimuth

The incidence of a source depends on a single parameter which is theazimuth θ. In these conditions, the directing vector a(θ) is a functionof θ. In the presence of M transmitters being propagated along P_(m)distributed multiple paths, the observation vector x(t) of the equation(4) becomes:

${x(t)} = {{\sum\limits_{m = 1}^{M}{\sum\limits_{p = 1}^{P_{m}}{x_{m\; p}(t)}}} + {b(t)}}$such that

$\begin{matrix}{{x_{m\; p}(t)} = {\int_{\theta_{m\; p} - {\delta\;\theta_{m\; p}}}^{\theta_{m\; p} + {\delta\;\theta_{m\; p}}}{{\rho(\theta)}{a(\theta)}{s_{m}\left( {t - {\tau(\theta)}} \right)}{\mathbb{e}}^{j\; 2\;\pi\;{f{(\theta)}}t}\ {\mathbb{d}\theta}}}} & (16)\end{matrix}$where θ_(mp) and δθ_(mp) respectively denote the center and the width ofthe diffusion cone associated with the pth multiple path of the mthtransmitter. The parameters τ(θ), f(θ) and ρ(θ) depend only on theazimuth θ of the diffuser. The equation (5) modeling a diffusion cone ofa source with L diffusers becomes:

$\begin{matrix}{{x_{m\; p}(t)} = {\sum\limits_{i = 1}^{L}{\rho_{i}{a\left( {\theta_{m\; p} + {\delta\;\theta_{mpi}}} \right)}{s_{m}\left( {t - \tau_{m\; p} - {\delta\;\tau_{mpi}}} \right)}{\mathbb{e}}^{j\; 2\;{\pi{({{fmp} + {\delta\;{fmpi}}})}}t}}}} & (17)\end{matrix}$1st Variant

FIG. 8 diagrammatically represents the steps of the method for thegoniometry of the distributed sources in azimuth.

The objective of the 1D goniometry of the distributed sources is todetermine the M_(T) doublets of parameters (θ_(mp), δθ_(mp)) whichminimize the criterion J_(diffusion)(θ, δθ). It is important to rememberthat δθ_(mp)=[δθ_(mp1) . . . δθ_(mpL)]^(T), bearing in mind that u^(T)denotes the transpose of u. According to the equations (11), (9) and(8), the criterion J_(diffusion)(θ, δθ) becomes:J _(diffusion)(θ,δθ)=det(Q ₁(θ,δθ))/det(Q ₂(θ,δθ)),  (18)with Q₁(θ, δθ)=D(θ, δθ)^(H) Π_(b) D(θ, δθ), ₂(θ, δθ)=D(θ, δθ)^(H) D(θ,δθ),and D(θ, δθ)=[a(θ+δθ₁) . . . a(θ+δθ_(L))]

FIG. 9 simulates the case of a distributed source of average incidenceθ₁₁=100° with a cone of width δθ₁₁=20° on a circular array with N=5sensors of radius R such that R/λ=0.8 (λ denotes the wavelength). InFIG. 9, the method is applied by breaking down the diffusion cone intoL=2 diffusers such that δθ=[Δθ−Δθ]^(T). In these conditions, thecriterion J_(diffusion)(θ, δθ) depends only on the two scalars θ and Δθ.In this FIG. 9, the function −10 log 10(J_(diffusion)(θ, Δθ)) isplotted, where the maxima correspond to the estimates of the parameterssought.

FIG. 9 shows that the method can well be used to find the center of thediffusion cone in θ₁₁=100° and that the cone going from incidence 80° to120° is broken down into two paths of incidences θ₁₁−Δθ₁₁=90° andθ₁₁+Δθ₁₁=110°. Bearing in mind that the parameter Δθ₁₁ reflects abarycentric distribution of the diffusers, it is possible to deduce fromthis that it is necessarily less than the width of the cone δθ₁₁.

FIG. 10, with the same array of sensors, simulates the case of twodistributed sources of average incidences θ₁₁=100° and θ₂₂=150° withcones of respective widths δθ₁₁=20° and δθ₂₂=5°. As in the case of thesimulation of FIG. 9, the goniometry is applied with L=2 diffusers whereδθ=[Δθ−Δθ]^(T).

FIG. 10 shows that the method can be used to estimate with accuracy thecenters of the diffusion cones θ₁₁ and θ₂₂ and the parameters Δθ₁₁ andΔθ₂₂ linked to the width of the diffusion cones such that the width ofthe cone satisfies: δθ_(mp)=2×Δθ_(mp).

FIG. 11 represents the steps of the symmetrical version of the variantdescribed in FIG. 8.

For a goniometry in azimuth, the solution (θ,δθ) necessarily leads tothe solution (θ,−δθ). From this comment, it is possible to deduce thefollowing two equations:c(θ,δθ,α)^(H) E _(b)=0,c(θ,−δθ,α)^(H) E _(b)=0,such that, according to (6) (8) (18):

$\begin{matrix}{{{c\left( {\theta,{\delta\;\theta},\alpha} \right)} = {{\sum\limits_{j = 1}^{L}{\alpha_{j}{a\left( {\theta + {\delta\;\theta_{j}}} \right)}}} = {{D\left( {\theta,{\delta\;\theta}} \right)}\alpha}}},} & \left( {18\text{-}1} \right)\end{matrix}$where the matrix E_(b) depends on the (N−K) eigenvectors e_(MT+i)(1≦i≦N−K) associated with the lowest natural values of the covariancematrix R_(xx)=E[x(t) x(t)^(H)], such that: E_(b)=[e_(K+1) . . . e_(N)].From the expression (18-1) it can be deduced from this that to estimatethe parameters (θ_(mp), δθ_(mp)), it is necessary to search for theminima that cancel the following pseudo-spectrum:

$\begin{matrix}{{{{J_{{MUSIC\_ diff}{\_ sym}}\left( {\theta,{\delta\;\theta},\alpha} \right)} = \frac{{c_{s}^{H}\left( {\theta,{\delta\;\theta},a} \right)}\Pi_{bs}{c_{s}\left( {\theta,{\delta\;\theta},\alpha} \right)}}{{c_{s}^{H}\left( {\theta,{\delta\;\theta},\alpha} \right)}{c_{s}\left( {\theta,{\delta\;\theta},\alpha} \right)}}},{{c_{s}\left( {\theta,{\delta\;\theta},\alpha} \right)} = {\begin{bmatrix}{c\left( {\theta,{\delta\;\theta},\alpha} \right)} \\{c\left( {\theta,{{- \delta}\;\theta},\alpha} \right)}\end{bmatrix}\mspace{14mu}{and}}}}{\Pi_{bs} = {\frac{1}{2}E_{bs}E_{bs}^{H}\mspace{14mu}{where}}}\text{}{E_{bs} = \begin{bmatrix}E_{b} \\E_{b}\end{bmatrix}}} & \left( {18\text{-}2} \right)\end{matrix}$According to the expressions (18-1) (18-2), the vector c_(s)(θ, δθ, α)can be written as follows:

$\begin{matrix}{{{{c_{s}\left( {\theta,{\delta\;\theta},\alpha} \right)} = {{D_{s}\left( {\theta,{\delta\;\theta}} \right)}\alpha}},{with}}{{{D_{s}\left( {\theta,{\delta\;\theta}} \right)} = \begin{bmatrix}{D\left( {\theta,{\delta\;\theta}} \right)} \\{D\left( {\theta,{{- \delta}\;\theta}} \right)}\end{bmatrix}},}} & \left( {18\text{-}3} \right)\end{matrix}$

The minimizing of J_(MUSIC) _(—) _(diff) _(—) _(sym)(θ, δθ, α) relativeto α will lead to the criterion J_(diffusion) _(—) _(sym)(θ, δθ). Toobtain J_(diffusion) _(—) _(sym)(θ,δθ), all that is needed is to replacein expression (18), D(θ,δθ) with D_(s)(θ,δθ) and Π_(b) with Π_(bs). Thefollowing is thus obtained:J _(diffusion-sym)(θ,δθ)=det(Q _(1s)(θ,δθ))/det(Q _(2s)(θ,δθ)),  (18-4)with Q_(1s)(θ, δθ)=D_(s)(θ, δθ)^(H) Π_(bs) D_(s)(θ, δθ) and Q_(2s)(θ,δθ)=D_(s)(θ,δθ)^(H) D_(s)(θ, δθ)

The M_(T) doublets of parameters (θ_(mp), δθ_(mp)) that minimize thecriterion J_(diffusion-sym)(θ,δθ) are therefore sought.

2nd Variant

FIG. 12 diagrammatically represents the steps of the second variant ofthe goniometry of the distributed sources in azimuth.

By performing a limited development of the order I of a(θ+δθ_(i)) aboutthe central incidence θ, the expression (13) becomes as follows:

$\begin{matrix}{{{a\left( {\theta + {\delta\;\theta_{i}}} \right)} = {{U(\theta)}{k\left( {\delta\;\theta_{i}} \right)}}}{where}{{U(\theta)} = \left\lbrack {{a(\theta)}\frac{\partial{a(\theta)}}{\partial\theta}\mspace{14mu}\ldots\mspace{14mu}\frac{\partial\left( {a(\theta)} \right)^{I}}{\partial\theta^{I}}} \right\rbrack}{and}{{k\left( {\delta\;\theta_{i}} \right)} = \begin{bmatrix}1 \\{\delta\;\theta_{i}} \\\vdots \\\frac{\delta\;\theta_{i}^{I}}{I!}\end{bmatrix}}} & (19)\end{matrix}$It can be deduced from this that the vector c(θ, δθ, α) of theexpression (18-1) is written, according to (14):

$\begin{matrix}{{{c\left( {\theta,{\delta\;\theta},\alpha} \right)} = {{U(\theta)}{\beta\left( {{\delta\;\theta},\alpha} \right)}}}{with}{{{\beta\left( {{\delta\;\theta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}{\alpha_{j}{k\left( {\delta\;\theta_{j}} \right)}}}},}} & \left( {19\text{-}1} \right)\end{matrix}$

The aim of the second variant of the 1D goniometry of diffuse sources isto determine the M_(T) incidences θ_(mp) that minimize the criterionJ_(diffusion) ^(opt)(θ). According to the equations (15) and (14), thecriterion J_(diffusion) ^(opt)(θ) becomes:J _(diffusion) ^(opt)(θ)=det(Q ₁ ^(opt)(θ))/det(Q ₂ ^(opt)(θ)),  (20)with Q₁ ^(opt)(θ)=U(θ)^(H) Π_(b) U(θ) and Q₂ ^(opt)(θ)=U(θ)^(H) U(θ),and

${{\beta\left( {{\delta\;\theta},\alpha} \right)} = {\sum\limits_{j = 1}^{L}{\alpha_{j}{k\left( {\delta\;\theta_{j}} \right)}}}},$

Determining the vectors δθ_(mp) entails estimating the vector δ(δθ_(mp),α): For this, all that is needed is to find the eigenvector associatedwith the minimum natural value of Q₂ ^(opt)(θ_(mp))⁻¹ Q₁ ^(opt)(θ_(mp)).

In FIG. 13, the MUSIC performance levels are compared with those of thesecond variant of distributed MUSIC for I=1 and I=2. The array ofsensors is that of FIG. 9 and of FIG. 10. The case of two distributedsources of average incidence θ₁₁=100° and θ₂₂=120° with cones ofrespective width δθ₁₁=20° and δθ₂₂=20° is simulated. It should beremembered that the M_(T) maxima of the function −10 log10(J_(diffusion) ^(opt)(θ)) are the estimates of the incidences θ_(mp)sought. The curves of FIG. 13 show that the more the order I of thelimited development increases, the higher the level of the two maxima ofthe criterion becomes, because there is a convergence towards a goodapproximation of the model. Table 1 gives the estimates of theincidences for the three methods.

TABLE 1 Goniometry in azimuth of a distributed source (θ₁₁ = 100° θ₂₂ =120° with a cone of width δθ₁₁ = 20° δθ₂₂ = 20°) with the second variant{circumflex over (θ)}₁₁ given that {circumflex over (θ)}₂₂ given thatθ₁₁ = 100° θ₂₂ = 120° Conventional 97 123.1 MUSIC (I = 0) Distributed101.7 118 MUSIC (I = 1) Distributed 99.2 120.6 MUSIC (I = 2)

Table 1 confirms that the lowest incidence estimation bias is obtainedfor I=2, that is, for the distributed MUSIC method of order ofinterpolation of the highest directing vector interpolation order. Thesimulation of FIG. 10 and of table 1 is obtained for a time spread ofthe two zero sources. More precisely, the delays δτ_(11i) and δτ_(22i)of (17) of the diffusers are zero. In the simulation of table 2 and ofFIG. 11, the preceding configuration is retained, but with a time spreadof one sampling period T_(e) introduced such that: max (δτ_(mmi))−min(δτ_(mmi))=T_(e)

TABLE 2 Goniometry in azimuth of partially correlated distributedsources θ₁₁ = 100° θ₂₂ = 120° with a cone of width δθ₁₁ = 20° δθ₂₂ = 20°with the 2nd variant {circumflex over (θ)}₁₁ given that {circumflex over(θ)}₂₂ given that θ₁₁ = 100° θ₂₂ = 120° Conventional 96.2 124.3 MUSIC (I= 0) Distributed 98.1 121.5 MUSIC (I = 2)

The results of Table 2 and of FIG. 14 show that the methods envisaged inthis invention take into account the configurations ofpartially-correlated diffusers

In this second variant it is possible, as in the first variant, to takeinto account the symmetry of the solutions in order to eliminate someambiguities. For this, it should first be noted that, according to (19)(19-1):

$\begin{matrix}{{{c\left( {\theta,{{- \delta}\;\theta},\alpha} \right)} = {{U_{1}(\theta)}{\beta\left( {{\delta\;\theta},\alpha} \right)}}}{with}{{U_{1}(\theta)} = \left\lbrack {{a(\theta)} - {\frac{\partial{a(\theta)}}{\partial\theta}\mspace{14mu}\ldots\mspace{14mu}\left( {- 1} \right)^{I}\frac{\partial\left( {a(\theta)} \right)^{I}}{\partial\theta^{I}}}} \right\rbrack}} & \left( {20\text{-}1} \right)\end{matrix}$According to (18-1) (18-2), a new expression of the vector c_(s)(θ, δθ,α) can be deduced:

$\begin{matrix}{{{c_{s}\left( {\theta,{\delta\;\theta},\alpha} \right)} = {{U_{s}(\theta)}{\beta\left( {{\delta\;\theta},\alpha} \right)}}}{and}{{U_{s}(\theta)} = \begin{bmatrix}{U\left( {\theta,\alpha} \right)} \\{U_{1}\left( {\theta,\alpha} \right)}\end{bmatrix}}} & \left( {20\text{-}2} \right)\end{matrix}$

By replacing, in the equation (20) U_(s)(θ) with U(θ) and Π_(b) withΠ_(bs), it can be deduced from this, according to (18-2) (18-4), that toestimate the M_(T) incidences (θ_(mp)), all that is needed is tominimize the following one-dimensional criterion:J _(diffusion) _(—) _(sym) ^(opt)(θ)=det(Q _(1s) ^(opt)(θ))/det(Q _(2s)^(opt)(θ)),  (20-3)with Q_(1s) ^(opt)(θ)=U_(s)(θ)^(H) Π_(bs) U_(s)(θ) and Q_(2s)^(opt)(θ)=U_(s)(θ)^(H) U_(s)(θ),

This symmetrical version of the second variant of the goniometry ofdistributed sources in azimuth is summarized in FIG. 15.

-   [1] RO. SCHMIDT “A signal subspace approach to multiple emitter    location and spectral estimation”, PhD thesis, Stanford University    CA, November 1981.-   [2] FERRARA, PARKS “Direction finding with an array of antennas    having diverse polarizations”, IEEE trans on antennas and    propagation, March 1983.-   [3] S. VALAE, B. CHAMPAGNE and P. KABAL “Parametric Localization of    Distributed Sources”, IEEE trans on signal processing, Vol 43 no 9    September 1995.-   [4] D. ASZTELY, B. OTTERSTEN and AL. SWINDLEHURST “A Generalized    array manifold model for local scattering in wireless    communications”, Proc of ICASSP, pp 4021-4024, Munich 1997.-   [5] M. BENGTSSON and B. OTTERSTEN “Low-Complexity Estimators for    Distributed Sources”, trans on signal processing, vol 48, no 8,    August 2000.

It will be readily seen by one of ordinary skill in the art that thepresent invention fulfils all of the objects set forth above. Afterreading the foregoing specification, one of ordinary skill in the artwill be able to affect various changes, substitutions of equivalents andvarious aspects of the invention as broadly disclosed herein. It istherefore intended that the protection granted hereon be limited only bydefinition contained in the appended claims and equivalents thereof.

1. A goniometry method of determining incidence parameters θ_(mp) andΔ_(mp) and deflection parameters δθ_(mp) and δΔ_(mp) of a diffusioncone, the diffusion cone being a model representing signals originatedfrom a transmitter via a path to an array of sensors, the methodcomprising: receiving, by the sensors, the signals originated from thetransmitter via the path; arranging the received signals according to anequation:${{x_{mp}(t)} = {\sum\limits_{i = 1}^{L}\;{\rho_{j}{a\left( {{\theta_{mp} + {\delta\;\theta_{mpi}}},{\Delta_{mp} + {\delta\;\Delta_{mpi}}}} \right)}{s_{m}\left( {t - \tau_{mp} - {\delta\;\tau_{mpi}}} \right)}{\mathbb{e}}^{{{j2\pi}{({{fmp} + {\delta\;{fmpi}}})}}t}}}},$the diffusion cone being represented by L diffusers, and the ithdiffuser having parameters (θ_(mp), δθ_(mpi), Δ_(mp), δΔ_(mpi)), wherea(θ_(mp)+δθ_(mpi), Δ_(mp)+δΔ_(mpi)) represent a directing vectorassociated with the ith diffuser, θ_(mp) is the incidence angle inazimuth for the transmitter, δθ_(mpi) is the offset in azimuthal angleof incidence corresponding to the ith diffuser, Δ_(mp) is the elevationangle of incidence for the transmitter, δΔ_(mpi) is the offset inelevational angle of incidence corresponding to the ith diffuser, τrepresents delays, f represents Doppler shifts, and ρ representsattenuation; deriving a vector D(θ, Δ, δθ, δΔ) α by combining thedirecting vectors associated with the L diffusers or deriving a vectorU(θ, Δ) β (δθ, δΔ, α) according to a directing vector a(θ, Δ) andderivatives of the directing vector a(θ, Δ), where α represents acombination vector associated with the L diffusers; applying aMUSIC-type goniometry algorithm based on the vector D(θ, Δ, δθ, δΔ) α orthe vector U(θ, Δ) β (δθ, δΔ, α) to determine at least one of theparameters θ_(mp), Δ_(mp), δθ_(mp), or δΔ_(mp).
 2. The goniometry methodas claimed in claim 1, wherein the MUSIC-type goniometry algorithmcomprises a minimizing step performed based on the vector D(θ, Δ, δθ,δΔ).
 3. The goniometry method as claimed in claim 2, wherein theMUSIC-type goniometry algorithm comprises a minimizing step performedbased on a vector D_(s)(θ, Δ, δθ, δΔ), which is derived from the vectorD(θ, Δ, δθ, δΔ), where the parameters δθ and/or δΔ in D(θ, Δ, δθ, δΔ)is/are replaced by additive inverse(s) thereof.
 4. The goniometry methodas claimed in claim 1, further comprising: separating the incidenceparameters and the deflection parameters; wherein the MUSIC-typegoniometry algorithm comprises a minimizing step performed according tothe parameters (θ, Δ) on a matrix U(θ, Δ) in order to determine theincidence parameters θ_(mp), Δ_(mp), then subsequently to determine thedeflection parameters δθ_(mp), δΔ_(mp) from the incidence parametersθ_(mp), Δ_(mp).
 5. The goniometry method as claimed in claim 4, whereinthe minimizing step is performed on the matrix U_(s)(θ, Δ) derived fromU(θ) by replacing the parameters δθ and/or δΔ with additive inverse(s)thereof.
 6. The goniometry method as claimed in claim 1, wherein theMUSIC-type goniometry algorithm comprises a minimizing step performedbased on the vector D(θ, δθ) which depends only on the azimuth angle θand on the deflection vector δθ corresponding to the azimuth angle θ. 7.The goniometry method as claimed in claim 6, wherein the minimizing stepis performed on the matrix D_(s)(θ, δθ), which is derived from thevector D(θ, δθ), where the parameter δθ in D(θ, δθ) is replaced byadditive an inverse thereof.
 8. The goniometry method as claimed inclaim 1, wherein the MUSIC-type goniometry algorithm comprises aminimizing step being performed based on a matrix U(θ), which dependsonly on the azimuth angle θ, in order to determine the incidenceparameter θ_(mp) and to subsequently determine the deflection parametersδθ_(mp).
 9. The goniometry method as claimed in claim 8, wherein theminimizing step is performed on the matrix U_(s)(θ) derived from U(θ) byreplacing the parameters δθ with additive inverse(s) thereof.